Common use of Combinatorial cuts Clause in Contracts

Combinatorial cuts. The optimal solution of SP(αˆ) provides the best assignment (v∗, q∗, c∗) associated with the contracts defined by αˆ. In particular, it defines a feasible solution of MDPC, and its value ZSP (αˆ) is an upper bound on the optimal value of MDPC. The incumbent or best-known value, i.e., Zinc, is compared with ZSP (αˆ) and is updated if needed, that is, if ZSP (αˆ) < Zinc. Moreover, a combinatorial cut of the following type can be added to the MP formulation: Σ , Σ αt + Σ (1 − αt ) ≥ 1. (16) t∈T e∈E:αˆt =0 e∈E:αˆt =1 This combinatorial cut removes the solution αˆ from the feasible space of MP. In other words, the cut expresses that the contract plan αˆ has already been handled, and therefore αˆ can be ruled out from the subsequent search process. by switching some value αˆt from 0 to 1 (i.e., by activating a new contract at time t) so as to satisfy The combinatorial cut (16) can be further strengthened by restricting the first double sum to those pairs (t, e) such that ΣHe−1 αˆt−n = 0. Indeed, suppose that a new contract plan is to be considered